Chapter 3 · Lesson 3 of 10 · Grade 7 · Expressions & Equations · MAT-07-EE-003
Factor Linear Expressions
Represent and solve factor linear expressions problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 3: Expressions, Equations, and Inequalities
Expressions & Equations
Essential question: How can equivalent expressions and equations model constraints and support justified solutions?
Where this lesson fits
Lesson 3 of 10. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Create and defend a multi-step model using the central ideas of expressions, equations, and inequalities, including a representation, calculation, interpretation, and independent check.
Learning objectives
What you should be able to do
- Represent and solve factor linear expressions problems using precise mathematical notation, models, and reasoning.
- Explain why a method for factor linear expressions works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
Grade 7 uses prerequisite knowledge as active reasoning.
Learn
Build the idea from meaning, not memorization.
Chapter 3: Expressions, Equations, and Inequalities
Expressions & Equations. Lesson 3 of 10 extends earlier chapter reasoning.
Connection to the course
Use earlier Grade 7 chapters, Grade 6 prerequisite knowledge as active knowledge. Connect today's representation to at least one earlier chapter representation.
Mathematical meaning
Factor Linear Expressions uses symbols to describe relationships. Equality and inequality are relationships that must be preserved on both sides.
Represent the relationship
The Math Box uses algebra tiles to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify algebra by substituting the proposed solution into the original statement; equivalent expressions must agree for the same input.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Algebra Tiles Lab
Model a linear expression with x-tiles and unit tiles, then evaluate it.
- Model factor linear expressions in the Math Box.
- Change one input and predict the effect before testing it.
- Connect the visual, numerical, and symbolic representations.
- State what remains invariant and why.
Worked examples
Twenty different ways to see the concept work.
Factor 4x + 8.
2(2x + 4)Factor out the greatest common factor.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Factor 9x + 15.
3(3x + 5)The representation should show the same mathematical relationship as the calculation. Factor out the greatest common factor.
Explain why a valid method works, then solve: Factor 16x + 24.
4(4x + 6)A complete explanation names the relationship or property being preserved. Factor out the greatest common factor.
Estimate or predict first, then calculate and decide whether the result is reasonable: Factor 25x + 35.
5(5x + 7)The estimate is a reasonableness check, not a replacement for the exact result. Factor out the greatest common factor.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Factor 36x + 48.
6(6x + 8)Verification should independently support the result. Factor out the greatest common factor.
Interpret the answer in context after solving. What does the result mean here? Factor 49x + 63.
7(7x + 9)State the result with its meaning, units, direction, or comparison—not only a number. Factor out the greatest common factor.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Factor 16x + 24.
Use equivalent operations on both sides. Correct solution: 8(2x + 3)Error analysis requires identifying the broken idea, not merely replacing the final answer. Factor out the greatest common factor.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Factor 6x + 8. Case B: Factor 24x + 24.
Case A: 2(3x + 4) Case B: 4(6x + 6)Compare the structure, representation, units, and result rather than only the numbers. Factor out the greatest common factor.
Factor 12x + 15.
3(4x + 5)Factor out the greatest common factor.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Factor 20x + 24.
4(5x + 6)The representation should show the same mathematical relationship as the calculation. Factor out the greatest common factor.
Explain why a valid method works, then solve: Factor 30x + 35.
5(6x + 7)A complete explanation names the relationship or property being preserved. Factor out the greatest common factor.
Estimate or predict first, then calculate and decide whether the result is reasonable: Factor 42x + 48.
6(7x + 8)The estimate is a reasonableness check, not a replacement for the exact result. Factor out the greatest common factor.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Factor 14x + 63.
7(2x + 9)Verification should independently support the result. Factor out the greatest common factor.
Interpret the answer in context after solving. What does the result mean here? Factor 24x + 24.
8(3x + 3)State the result with its meaning, units, direction, or comparison—not only a number. Factor out the greatest common factor.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Factor 8x + 8.
Use equivalent operations on both sides. Correct solution: 2(4x + 4)Error analysis requires identifying the broken idea, not merely replacing the final answer. Factor out the greatest common factor.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Factor 15x + 15. Case B: Factor 10x + 35.
Case A: 3(5x + 5) Case B: 5(2x + 7)Compare the structure, representation, units, and result rather than only the numbers. Factor out the greatest common factor.
Factor 24x + 24.
4(6x + 6)Factor out the greatest common factor.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Factor 35x + 35.
5(7x + 7)The representation should show the same mathematical relationship as the calculation. Factor out the greatest common factor.
Explain why a valid method works, then solve: Factor 12x + 48.
6(2x + 8)A complete explanation names the relationship or property being preserved. Factor out the greatest common factor.
Estimate or predict first, then calculate and decide whether the result is reasonable: Factor 21x + 63.
7(3x + 9)The estimate is a reasonableness check, not a replacement for the exact result. Factor out the greatest common factor.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Factor 32x + 24.
Hint: Choose a representation before computing.
Answer: 8(4x + 3)
02Interpret the answer in context after solving. What does the result mean here? Factor 10x + 8.
Hint: Explain why the method works and name a verification strategy.
Answer: 2(5x + 4)
03Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Factor 18x + 15.
Hint: Choose a representation before computing.
Answer: Use equivalent operations on both sides. Correct solution: 3(6x + 5)
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Factor 28x + 24. Case B: Factor 24x + 48.
Hint: Explain why the method works and name a verification strategy.
Answer: Case A: 4(7x + 6) Case B: 6(4x + 8)
05Factor 10x + 35.
Hint: Choose a representation before computing.
Answer: 5(2x + 7)
06Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Factor 18x + 48.
Hint: Explain why the method works and name a verification strategy.
Answer: 6(3x + 8)
Practice
Independent practice
01Explain why a valid method works, then solve: Factor 28x + 63.
Answer: 7(4x + 9)
02Estimate or predict first, then calculate and decide whether the result is reasonable: Factor 40x + 24.
Answer: 8(5x + 3)
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Factor 12x + 8.
Answer: 2(6x + 4)
04Interpret the answer in context after solving. What does the result mean here? Factor 21x + 15.
Answer: 3(7x + 5)
05Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Factor 8x + 24.
Answer: Use equivalent operations on both sides. Correct solution: 4(2x + 6)
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Factor 15x + 35. Case B: Factor 42x + 63.
Answer: Case A: 5(3x + 7) Case B: 7(6x + 9)
07Factor 24x + 48.
Answer: 6(4x + 8)
08Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Factor 35x + 63.
Answer: 7(5x + 9)
Common mistakes
Learn to catch the error, not just the answer.
Use equivalent operations on both sides.
Only terms with the same variable part are like terms.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Translate cost, distance, rate, and measurement situations into symbolic rules.
- Use equations and inequalities to determine unknown quantities, thresholds, and feasible ranges.
Challenge problems
Explain why a valid method works, then solve: Factor 48x + 24.
8(6x + 3)
Estimate or predict first, then calculate and decide whether the result is reasonable: Factor 14x + 8.
2(7x + 4)
Solve and verify the result with a second method, inverse operation, or equivalent representation: Factor 6x + 15.
3(2x + 5)
Interpret the answer in context after solving. What does the result mean here? Factor 12x + 24.
4(3x + 6)
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Chapter 3 Reasoning Lab
Solve factor linear expressions tasks using calculation, representation, explanation, interpretation, verification, error analysis, and comparison.
Practice
Mastery check
01Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Factor 20x + 35.
Answer: Use equivalent operations on both sides. Correct solution: 5(4x + 7)
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Factor 30x + 48. Case B: Factor 16x + 24.
Answer: Case A: 6(5x + 8) Case B: 8(2x + 3)
03Factor 42x + 63.
Answer: 7(6x + 9)
04Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Factor 56x + 24.
Answer: 8(7x + 3)
05Explain why a valid method works, then solve: Factor 4x + 8.
Answer: 2(2x + 4)
06Estimate or predict first, then calculate and decide whether the result is reasonable: Factor 9x + 15.
Answer: 3(3x + 5)
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Factor 16x + 24.
Answer: 4(4x + 6)
08Interpret the answer in context after solving. What does the result mean here? Factor 25x + 35.
Answer: 5(5x + 7)
Terminology
Words to know
- variable
- A symbol representing a number that may vary or be unknown.
- coefficient
- A numerical factor multiplying a variable.
- expression
- A combination of numbers, variables, and operations.
- solution
- A value that makes an equation or inequality true.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 7 scope, sequence, and public task structure
Coherent sequencing, representations, and reasoning-rich tasks.Eureka Math² Grade 7 program structure
Module coherence, cumulative practice, and fluency.enVision Mathematics Grade 7 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 7
Grade-level content and mathematical-practice expectations.